Question: Fatima conducts emissions inspections on cars. She finds that $6\%$ of the cars fail the inspection. Let $C$ be the number of cars Fatima inspects until a car fails an inspection. Assume that the results of each inspection are independent. Find the probability that the first failed inspection occurs on Fatima's $5^{\text{th}}$ inspection. You may round your answer to the nearest hundredth. $P(C=5)=$
Solution: Without a fancy calculator For each inspection: $P({\text{fail}})=0.06$ $P(\text{pass}})=0.94$ If the first failed inspection occurs on the $5^{\text{th}}$ inspection, then Fatima's sequence of results needs to be "pass, pass, pass, pass, fail." $\begin{aligned} P(C=5)&=P(\text{pass}}, \text{pass}}, \text{pass}}, \text{pass}}, {\text{fail}}) \\\\ &=(0.94})(0.94})(0.94})(0.94})({0.06}) \\\\ &=(0.94)^4(0.06) \\\\ &\approx0.0468 \end{aligned}$ $P(C=5)\approx 0.0468\approx0.05$